The treatment of Backlund transformations (BTS) in multi-dimensional spaces via gauge transformations is presented. The general two-dimensional N*N matrix Zakharov-Shabat-AKNS spectral problem and the general two-dimensional Nth-order Zakharov-Shabat-Gelfand-Dikij (ZSGD) spectral problem are considered in detail. The structure of the infinite-dimensional group of general BTS is studied. It is shown that one has N elementary BTS for the N*N matrix problem and only one elementary BT for the Nth-order ZSGD problem. The explicit formulae for soliton and elementary BTS for both spectral problems are found. Non-linear superposition formulae for elementary BTS for both problems are obtained. An infinite lattice of solutions of the integrable evolution equations related to the N*N matrix problem is constructed.
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Boiti et al. (1985) studied this question.
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